Control of High-Order Systems Using Simple Models.

Transcription

1 Luisiana State University LSU Digital Cmmns LSU Histrical Dissertatins and Theses Graduate Schl 1971 Cntrl f High-Order Systems Using Simple Mdels. Rbert Andrew Mllenkamp Luisiana State University and Agricultural & Mechanical Cllege Fllw this and additinal wrks at: Recmmended Citatin Mllenkamp, Rbert Andrew, "Cntrl f High-Order Systems Using Simple Mdels." (1971). LSU Histrical Dissertatins and Theses This Dissertatin is brught t yu fr free and pen access by the Graduate Schl at LSU Digital Cmmns. It has been accepted fr inclusin in LSU Histrical Dissertatins and Theses by an authrized administratr f LSU Digital Cmmns. Fr mre infrmatin, please cntact gradetd@lsu.edu.

2 71-20,610 MOLLENKAMF, Rbert Andrew, CONTROL OF HIGH-ORDER SYSTEMS USING SIMPLE MODELS. The Luisiana State University and Agricultural and Mechanical Cllege, Ph.D., 1971 Engineering, chemical U niversity Micrfilms, A XEROX Cmpany, Ann Arbr, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED

3 CONTROL OF HIGH-ORDER SYSTEMS USING SIMPLE MODELS A Dissertatin Submitted t the Graduate Faculty f the Luisiana State University and Agricultural and Mechanical Cllege in partial fulfillment f the requirements fr the degree f Dctr f Philsphy The Department f Chemical Engineering in by Rbert And rew Mllenkamp B.S., University f Missuri, 1964 M.S., University f Missuri, 1968 January, 1971

4 PLEASE NOTE: Sme pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS.

5 ACKNOWLEDGEMENT This wrk was cnducted under the directin f Dr. Cecil L. Smith, J r., Assciate Prfessr f Chemical Engineering at Luisiana State University. The authr extends his appreciatin t Dr, Smith fr the many hurs f cnsultatin and guidance. His friendly and helpful manner did much t make this wrk thrughly enjyable as well as educatinal. The authr wishes t express his appreciatin fr the supprt f this wrk by Prject THEMIS Cntract N. F C-0021 administered fr the U. S. Department f Defense by the U. S. Air Frce Office f Scientific Research, Thanks are als extended t the Ethyl Crpratin whse fellwship supprted the authr fr the past year. The authr als thanks the Dr. Charles E. Cates Memrial Fund, dnated by Gerge H. Cates, fr prviding financial assistance fr the preparatin f this manuscript.

6 TABLE OF CONTENTS Page ACKNOWLEDGEMENT LIST OF FIGURES LIST OF TABLES ABSTRACT VI xiv xv CHAPTER I II III INTRODUCTION TUNING OF CONTINUOUS CONTROLLERS FOR HIGH ORDER SYSTEMS Intrductin Heat Exchanger with Temperature Cntrl Cntrller Tuning Tuning Using an Accurate Simulatin First-Order Plus Dead Time Apprximatin Optimal Cntrl Tuning Methd fr Lad Changes Mdificatin fr Setpint Changes Cmparisn f Tuning Results Summary Nmenclature Literature Cited TUNING OF SAMPLED-DATA CONTROLLERS FOR HIGH- ORDER SYSTEMS iil

7 Intrductin 40 Sampled-Data Temperature Cntrl f a Heat Exchanger 41 Digital PI and PID Cntrller Tuning 42 Deadbeat Algrithm fr Setpint Changes 43 Digital Predictr Algrithm 46 Cmparisn f Tuning Results 51 Evaluatin f Cntrller Algrithms 67 Summary 72 Nmenclature 76 Literature Cited 77 IV ON-LINE MODEL IDENTIFICATION AN D CONTROL USING THE KALMAN FILTER 78 Intrductin 78 Kalman Filtering fr Stchastic Prcesses 79 Parameter and Disturbance Identificatin 83 Kalman Filtering f First- Order Prcesses 83 Kalman Filtering f Secnd-Order Prcesses 86 First-Order Filter Perfrmance 89 Secnd-Order Filter Perfrmance 94 Summary 116 Nmenclature 125 Literature Cited 126 V SIMPLE MODEL IDENTIFICATION A ND CONTROL OF HIGH- ORDER PROCESSES USING THE KALMAN FILTER 127 Intrductin 127 iv

8 IV APPENDIX A B VITA First-Order Mdel Identificatin Secnd-Order Mdel Identificatin Mdel Feedback Cntrl First-Order Mdel Cntrl Secnd-Order Mdel Cntrl Summary Nmenclature Literature Cited CONCLUSION HEAT EXCHANGER SIMULATION FOR DETERMINING OPTIMAL CONTROLLER SETTINGS KALMAN FILTER IDENTIFICATION AND CONTROL PROGRAM FOR SECOND-ORDER PROCESS KALMAN FILTER PROGRAM FOR IDENTIFICATION OF FIRST- OR SECOND-ORDER MODEL OF A HEAT EXCHANGER v

9 LIST OF FIGURES Figure 2.1a 2.1b a 2.4b 2.5a 2.5b Schematic Diagram f Oil-Cndensing Steam Heat Exchanger Blck Diagram f Heat Exchanger with Feedback Temperature Cntrller Schematic Diagram f Heat Exchanger with Tube Wall and Tube Fluid Divided int Thermally Well-Mixed Sectins First-Order Apprximatin t the Heat Exchanger Prcess Reactin Curve Obtained by Making a 5 psi Change in the Steam Pressure '* Blck Diagram f Secnd-Order System and Disturbance, gq Blck Diagram f System Transfrmed t Third Order with Disturbance as Initial Cnditin Clsed-Lp Outlet Temperature Respnse t a Lad Change fr Different Values f r Cntrller Output Respnse t a Lad Change fr Different Values f r Optimal PI Respnses t Lad Change with IAE and ITAE Criteria Clsed-Lp PI Respnses t Lad Change with Optimal IAE and lst-order Plus Dead Time Parameters Clsed-Lp Respnses t Lad Change with Optimal ITAE and lst-order Plus Dead Time Parameters Page vi

10 Figure , Clsed-Lp PI Respnses t Setpint Change with Optimal IAE and lst-order Plus Dead Time Parameters Clsed-Lp PI Respnses t Setpint Change with Optimal ITAE and lst-order Plus Dead Time Parameters Clsed-Lp PID Respnses t Lad Change with Optimal IAE and lst-order Plus Dead Time Parameters Clsed Lp PID Respnses t Lad Changes with Optimal ITAE, lst-order Plus Dead Time, and Optimal Cntrl Parameters Clsed-Lp PID Respnses t Setpint Changes with Optimal ITAE, lst-order Plus Dead Time and Optimal Cntrl Parameters Clsed-Lp Respnses t Setpint Changes Using Mdified PID Algrithm with Optimal ITAE and Optimal Cntrl Methd Parameters Schematic Diagram f Sampled-Data Feedback Cntrl System Prcess Output, Cntrl Actin, and Calculated Disturbance f Digital Predictr Algrithm Clsed-Lp PI Respnses t Lad Changes with Optimal IAE and First-Order Plus Dead Time Parameters Clsed-Lp PI Respnses t Lad Changes with Optimal ITAE and First-Order Plus Dead Time Parameters Clsed-Lp PI Respnses t Setpint Changes with Optimal IAE and lst-order Plus Dead Time Parameters Clsed-Lp PI Respnses t Setpint Changes with Optimal ITAE and First-Order Plus Dead Time Parameters Page vii

11 Figure Clsed-Lp PID Respnses t Lad Changes with Optimal IAE and First-Order Plus Dead Time Parameters Clsed-Lp PID Respnses t Lad Changes with Optimal ITAE and First-Order Plus Dead Time Parameters Clsed-Lp PID Respnses t Setpint Changes with Optimal IAE and First-Order Plus Dead Time Parameters Clsed-Lp PID Respnses t Setpint Changes with Optimal ITAE and First-Order Plus Dead Time Parameters Clsed-Lp PI Respnses t Setpint Changes Using Optimal and First-Order Plud Dead Time ITAE Parameters frm Lad Tuning Clsed-Lp PID Respnses t Setpint Changes Using Optimal and First-Order Plus Dead Time ITAE Parameters frm Lad Tuning Clsed-Lp PI and PID Respnses t Lad Changes Using Optimal and First-Order Plus Dead Time ITAE Parameters frm Setpint Tuning Clsed-Lp Lad Change Respnses with Optimal ITAE, PI, and PID Algrithms and with Deadbeat Algrithm Clsed-Lp Setpint Change Respnses with Optimal ITAE, PI, and PID Algrithms and with Deadbeat Algrithm Clsed-Lp Lad Change Respnses with Optimal ITAE, PID, Deadbeat, and Optimal Deadbeat Clsed-Lp Lad Change Respnses with Optimal ITAE, PI and PID and with Predictr Using *3K Clsed-Lp Setpint Change Respnse with Optimal, ITAE, PI and PID and with Predictr Algrithm Using Page viii

12 Figure a 4.2b 4.2c 4.2d 4.3a 4.3b 4.3c 4.3d 4.4a 4.4b 4,4c 4,4d 4.5a Blck Diagram f Linear Stchastic Prcess with Kalman Filter fr State Estimatin Clsed-Lp, Nise-Free First-Order Prcess Output, C., and Kalman Filter Output Estimate,LX(1) Kalman Filter Estimates, X(2) and X( 3), f First-Order Prcess Parameters, a^ and a2 Kalman Filter Estimate, X(4), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, First-Order Prcess Output, C., and Kalman Filter Output Estimate, X t l ), with = Kalman Filter Estimates, X(2) and X(3), f First-Order Prcess Parameters, and a2 Kalman Filter Estimate, X(4), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, First-Order Prcess Output, C., and Kalman Filter Output Estimate, XC1), with = 100 Kalman Filter Estimates, X(2) and X( 3), f First-Order Prcess Parameters, and a2 Kalman Filter Estimate, X(4), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, First-Order Prcess Output, C., and Kalman Filter Output Estimate, Xtl), with \ = 20 Page ix

13 Figure 4.5b 4. 5c 4.5d 4.6a 4.6b 4.6c 4. 6d 4, 7a 4.7b 4.7c 4.7d 4.8a 4.8b 4.8c Kalman Filter Estimates, X(2) and X(3), f First-Order Prcess Parameters, a and a^ Kalman Filter Estimate, X(4), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, Nise-Free, Secnd-Order Prcess Output, C., and Kalman Filter Output Estimate, &(1) Kalman Filter Estimates X(2), X(3), X( 4), and X(5) f Secnd-Order Prcess Parameters a^, a^, a^ and Kalman Filter Estimate, X(6), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, Secnd-Order Prcess Output, C., and Kalman Filter Output Estimate, x h ), with = 0.25 Kalman Filter Estimates X(2), X( 3), X( 4), and X(5) f Secnd-Order Prcess Parameters a l> a2 > a3» an(^ Kalman Filter Estimate, X ( 6), f Actual Prcess Disturbance, d Manipulated Variable Using Kalman Filter Mdel fr Feedback Cntrl Clsed-Lp, Secnd-Order Prcess Output, C., and Kalman Filter Output Estimate, x h ), with Rj. = 20 Kalman Filter Estimates X(2), X(3), X(4), and X(5) f Secnd-Order Prcess Parameters a^, a^ * a3 > and Kalman Filter Estimate, X ( 6), f Actual Prcess Disturbance, d Page x

14 Figure 4.8d 4.9a 4.9b 4.9c 4.9d 5.1 Manipulated Variable Using Kalman Filter 120 Mdel fr Feedback Cntrl Clsed-Lp, Secnd-Order Prcess Output, 121 C,, and Kalman Filter Output Estimate, XC1), with RR = 1.0 Kalman Filter Estimates X(2), X(3), X(4), 122 and X(5) f Secnd-Order Prcess Parameters 1 > and a^ Kalman Filter Estimate, X( 6) f Actual 123 Prcess Disturbance, d Manipulated Variable Using Kalman Filter 124 Mdel fr Feedback Cntrl Blck Diagram f Heat Exchanger with Discrete 129 Feedback Cntrller and Kalman Filter Identificatin Page ' 5.2a 5.2 b 5.2c 5.3a 5.3b 5,3c 5.4a PI Cntrlled Setpi it Change Respnses f Heat Exchanger Outlet Temperature and Kalman Filter First-Order Mdel Estimate, X(l) Kalman Filter First-Order Mdel Parameter Estimates, X(2) and X(3) Kalman Filter First-Order Mdel Disturbance Estimate, X(4) PI Cntrlled Setpint Change Respnses f Heat Exchanger Outlet Temperature and Kalman Filter Secnd-Order Mdel Estimate, X(l) Kalman Filter Secnd-Order Mdel Parameter Estimates, X(2), X(3), X(4), and X(5) Kalman Filter Secnd-Order Mdel Disturbance Estimate, X( 6) Kalman Filter First-Order, Disturbance-Free Mdel Cntrlled Setpint Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) b 5.4c Kalman Filter First-Order Mdel Parameter 142 Estimates, X(2) and X(3) Kalman Filter First-Order Mdel Filter 143 Estimate, X(4) xi

15 Figure 5.5a 5.5b 5.5c 5.6a 5.6b 5.6c 5.7a 5.7b 5.7c 5.8i 5.9a 5.9b 5.9c 5.10a Kalman Filter First-Order Mdel Cntrlled Setpint Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) Kalman Filter First-Order Mdel Parameter Estimates, X(2) and X(3) Kalman Filter First-Order Mdel Disturbance Estimate, X(4) Kalman Filter First-Order Mdel Cntrlled Setpint and Lad Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) Kalman Filter First-Order Mdel Parameter Estimates, X(2) and X(3) Kalman Filter First-Order Mdel Disturbance Estimate, X(4) Kalman Filter First-Order Mdel Cntrlled Setpint Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) Kalman Filter First-Order Mdel Parameter Estimates, X(2) and X(3) Kalman Filter First-Order Mdel Disturbance Estimate, X(4) First-Order, Cnstant Parameter Mdel Cntrlled Setpint Change Respnse f Heat Exchanger Outlet Temperature Kalman Filter Secnd Order Mdel Cntrlled Setpint Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) Kalman Filter Secnd-Order Mdel Parameter Estimates, X ( 2 ), X ( 3 ), X ( 4 ), and X(5) Kalman Filter Secnd-Order Mdel Disturbance Estimate, X ( 6) Kalman Filter Secnd-Order Mdel Cntrlled Setpint and Lad Change Respnses f Heat Exchanger Outlet Temperature and Filter Estimate, X(l) Page xii

16 Figure Page 5.10b Kalman Filter Secnd-Order Mdel Parameter ig Estimates, X(2), X ( 3 ), X(4),X(5) 5.10c Kalman Filter Secnd-Order Mdel Disturbance igi Estimate, X(6) xiii

17 LIST OF TABLES Table Page 2.1 Results f Integral Tuning fr Lad Changes Results f Integral Tuning fr Setpint 25 Changes 3.1 Results f Integral Tuning fr Lad Changes Results f Integral Tuning fr Setpint 54 Changes xiv

18 ABSTRACT In this wrk the emphasis is n using simple mdels fr the cntrl f high-rder prcesses. In Chapter II this apprach is fllwed t tune cntinuus PI and PID cntrl algrithms. It was fund that PID cntrller settings determined by applying ptimal cntrl thery t a secnd-rder mdel f a heat exchanger gave nearly ptimal heat exchanger lad change respnse. Using the ptimal settings fr a first-rder lag plus dead time mdel, n the ther hand, gave pr heat exchanger respnses since n attempt was made t allw fr mdeling errr. Mdificatin f the PID algrithm resulted in an algrithm which gave nearly ptimal setpint change respnse using the ptimal lad change parameters. In Chapter III the tuning f discrete cntrllers by using first-rder plus dead time mdels was investigated. In additin t PI and PID cntrllers, attentin was given t a deadbeat and a predictr algrithm. Errr in mdeling the dynamics f a zer- rder hld in the discrete cntrl system necessitated altering the PI, PID, and predictr algrithm settings t achieve gd cntrl perfrmance. The zer-rder hld dynamics were nt apprximated with the deadbeat algrithm s that the deadbeat settings required n adjustment. xv

19 In Chapter IV the extended Kalman filter was applied t the identificatin and cntrl f first- and secnd-rder discrete prcesses. The Kalman filter mdels were utilized in a deadbeat cntrl algrithm which gave excellent setpint respnses. The filter was als able t reject nise which crrupted the measurement signal and still maintain gd cntrl perfrmance. In Chapter V the Kalman filter was used fr identifying first- and secnd-rder mdels f a heat exchanger. These mdels were used in a discrete cntrl algrithm which attempts t achieve a setpint respnse like the step respnse f a first-rder prcess. One disadvantage f the Kalman filter, hwever, is that several parameters must be specified t tune the filter and n easy methd is available t select these parameters t give ptimum filter p e r frmance. The resulting cntrl system prved able t adapt t changing prcess cnditins and still maintain excellent cntrl perfrmance. In this chapter, as well as the entire dissertatin, the need fr careful chice f the prcess mdel is demnstrated. xvi

20 CHAPTER I INTRODUCTION The cntrl f industrial prcesses is ften made mre difficult by the cmplexity f the prcess itself. A detailed mathematical descriptin f such cmplicated prcesses requires using high-rder r even partial differential equatins. The ecnmics f mst p r cesses, hwever, des nt justify such a detailed and specialized cntrl system design. Als, the implementatin f cmplex cntrl schemes is limited by the inflexibility f cntrl hardware fr a n a lg cntrl systems and by the availability f cmputatin time in digital cntrl systems. These prblems and restrictins have led t attempts t develp generalized and easily implemented cntrl strategies fr high-rder prcesses based n the behavir f mathematically simple prcess mdels. This dissertatin cnsiders the develpment and applicatin f such cntrl schemes fr bth cntinuus and discrete cntrl systems. Chapter II deals with IAE and ITAE integral criteria tuning f cntinuus feedback cntrllers. Tuning methds based n the behavir f simple mdels were applied t tuning a heat exchanger temperature cntrl system fr setpint and lad changes. One methd uses the 1

21 ptimal PI and PID settings fr a first-rder lag plus dead time mdel f the heat exchanger. Anther methd utilizes a secnd- rder mdel and the matrix Riccati equatin f ptimal cntrl thery t determine the ptimal lad change settings fr a PID cntrller. A mdified PID algrithm which nearly eliminates the dynamic differences between setpint and lad change respnses is als applied t cntrl f the heat exchanger temperature. In each case results f the tuning methd are cmpared with the actual ptimum respnses as determined with an accurate simulatin f the heat exchanger. Integral criteria tuning methds based n first-rder lag plus dead time mdels are applied in Chapter III t the tuning f discrete cntrllers. The first-rder lag plus dead time mdel methd fr cntinuus cntrl systems is mdified fr use with sampled-data PI and PID cntrllers. In additin t PI and PID cntrllers a minimal r deadbeat cntrl algrithm is evaluated. Als, a predictr algrithm is applied t heat exchanger temperature cntrl. This latter algrithm eliminates the effects f transprtatin lags in the prcess and cntrl hardware. Once again, an accurate simulatin f the heat exchanger and cntrl system prvides a means fr gauging the perfrmance f each tuning methd. The cntrl methds f Chapters II and III are based n simple mdels determined ff-line frm the prcess. Chapter IV, n the ther hand, deals with discrete n-line identificatin f first- and secnd-rder prcesses using the extended Kalman filter. The

22 3 n-line identificatin prvides a first- r secnd-rder mdel which is autmatically adjusted t fit changing prcess cnditins. The Kalman filter is als able t identify unmeasured prcess d i s turbances and t eliminate nise which crrupts the measurement f prcess variables. The extended Kalman filter is applied in Chapter V t determining first- and secnd-rder mdels f a heat exchanger and identifying unmeasured disturbances. The mdel is used t cntinually update a discrete feedback cntrl algrithm. The algrithm used is designed t give a clsed-lp setpint change respnse like the pen-lp step respnse f a first-rder prcess. Thus, this dissertatin examines the develpment and applicatin f cntrl systems, based n simple mdels, t high-rder prcesses. Bth cntinuus cntrl systems and discrete cntrl schemes are cnsidered.

23 CHAPTER II TUNING OF CONTINUOUS CONTROLLERS FOR HIGH ORDER SYSTEMS Intrductin A prblem f utmst imprtance in applying feedback cntrl t industrial prcesses is the tuning f the cntrller. Imprper cntrller tuning can result in expensive amunts f ff-specificatin prduct. It is fr this reasn that cntrller tuning has received cnsiderable attentin in the past. Hwever, the tuning techniques presently practiced are ultimately based n the behavir f first- and secnd-rder systems. Direct applicatin f such methds t high-rder systems culd result in unsatisfactry cntrl. The purpse f this study was t evaluate the relative merits f tuning techniques when applied t a typical high-rder system. Heat Exchanger with Temperature Cntrl A typical and mst imprtant high-rder system is the il- cndensing steam heat exchanger illustrated in Figure 2.1a. Oil flwing thrugh the heat exchanger tubes is heated by the steam cndensing n the shell side. The temperatures f the il and tube wall vary with bth time and psitin. The steam temperature, n the ther hand, is the same at any pint in the exchanger and, therefre, depends nly n time. Such a prcess, with tw independent variables, is termed a distributed-parameter system and, in general, is described by partial differential equatins.

24 steam, T s TRC V Y cndensate Figure 2.1-a. Schematic Diagram f Oil-Cndensing Steam Heat Exchanger setpint + TRC T=f(P) Prcess Figure 2.1-b. Blck Diagram f Heat Exchanger with Feedback Temperature Cntrller.

25 Cntrl f the utlet il temperature can be accmplished with a feedback cntrller as indicated in Figure 2.1a and the blck diagram f Figure 2.1b. The utlet il temperature is cntinuusly measured and fed back t the cntrller which adjusts the steam pressure until the utlet il temperature is at its setpint value. By far the mst cmmn types f feedback cntrllers used in the chemical prcess industry are the standard PI and PID cntrllers. The peratin f the PID, r prprtinal-integral-derivative, cntrller can be described by the cntrl algrithm: p = Kc [e + i-.dt + T d + pm (2.1) where: p = cntrller utput p m e = manual mde utput = cntrller errr-setpint-measurement K c = prprtinal gain T^ = integral time T^ = derivative time The PI, r prprtinal-integral, cntrl algrithm is btained by simply deleting the derivative term frm the PID algrithm. Thus, the tuning prblem is t select and T^ fr the PI c n trller r K, T., and T, fr the PID cntrller s that the c* l* d clsed-lp respnse f the utlet il temperature is ptimized in sme way. It is imprtant t nte that using the standard PID algrithm there is a basic difference between the dynamics f the setpint

26 7 and the lad change respnses. Examinatin.f Figure 2,1b and Equatin 2.1 indicates that the mment a setpint change ccurs, there is an immediate change in the errr and therefre, due t the prprtinal and derivative mdes, the cntrller utput. Hwever, when a lad change ccurs, the upset is dampened and delayed by the prcess. Fr this reasn, the prblem f ptimal cntrller tuning is twfld tuning fr setpint changes and tuning fr lad changes. Cntrller Tuning Perhaps the mst well knwn tuning techniques are the Ziegler- Nichls, Chen-Cn, and 3C methds (1). All f these attempt t achieve a decay rati f 1/4 fr.the clsed-lp temperature r e spnse, where the decay rati is defined as the rati f the versht n the secnd peak t the versht n the first peak. In general, hwever, there may be an infinite number f cmbinatins f cntrller tuning parameters which will give a decay rati equal t 1/4. The limitatins f 1/4-decay methds have led t the develpment f mre cnsistent tuning criteria--the integral criteria (2). With integral tuning methds the prblem is t determine the tuning parameters that minimize the cst functins: (2.2) CO (2.3)

27 8 (2.4) As ppsed t the 1/4-decay criterin, the integral criteria yield a unique set f tuning parameters. The ISE criterin was nt cnsidered in this study since it ften results in an scillatry respnse generally nt desirable fr "gd" prcess cntrl. Tuning Using an Accurate Simulatin Fr the temperature cntrlled heat exchanger the set f tuning parameters that minimize the integral criteria can be determined using an accurate dynamic mdel f the heat exchanger and cntrller. Fr a chsen upset, either setpint r lad, an initial best estimate is made f the tuning parameters. Next, the time respnse f the utlet fluid temperature is calculated, and the value f the chsen integral criterin is evaluated. The prcedure is repeated in a systematic way until the set f parameters which minimizes the integral is fund. The behavir f the heat exchanger is described by energy balances fr the il and the tube wall (3). Assuming plug flw f il in the tube and neglecting cnductin in the axial directin: (2.5) (2.6)

28 where: T = il temperature T w = wall temperature T g = steam temperature v = il velcity D^, Dq = inside and utside tube diameters h., h = inside and utside heat transfer cefficients i A., A = inside and utside tube areas available t heat i transfer V V -, = il and wall vlumes il wall P0 ^ > PWall = an<^ w a ^ C, C = il and wall heat capacities p il "il K2 U = P i l roll K 212 = h id itt/acpwallpwall K 322 =. pwall rwall Since the steam is saturated, the steam temperature is related t the steam pressure by the Clausius-Clapeyrn Equatin: wr =K<r-l> <2-7) s s s where: pg = steam pressure ps= steam pressure at reference state Tg = steam temperature Tg= steam temperature at reference state R = ideal gas law cnstant \ «latent heat f vaprizatin

29 The slutin f these equatins can be apprximated by dividing the tube side fluid and tube wall int sectins alng its length as in Figure 2.2. Assuming equal temperature at any pint in a given sectin, let: T - T as. = -i Jbi (2 8) ax l { J where: T. ^ = il temperature in sectin j-1 Tj = il temperature in sectin j L = length f a sectin With this mdificatin, the energy balances fr any sectin, becme: dt. v T j-1> + K 211 (Twj ' V <2 9) dt. i f - K322 <T s - V K212 <T«j - T J> (2-10> Using ten sectins fr the il and ten sectins fr the tube wall, the twenty rdinary differential equatins and feedback cntrller equatin were slved numerically t yield the clsed-lp time respnse. cupled with A furth-rder Runge-Kutta integratin rutine was a Pattern search prgram (4) t determine the set f tuning parameters which minimized a chsen integral criterin. Values fr the physical cnstants were taken frm an example heat exchanger calculatin by Kern (5) and are included in Appendix A w ith the digital cmputer prgram fr the heat exchanger.

30 shell tube wall 'j+1 tube il Figure 2.2. Schematic Diagram f Heat Exchanger with Tube Wall and Tube Fluid Divided int Thermally Well-Mixed Sectins.

31 Obviusly, this prcedure is t cmplex and time cnsuming t be practical fr cntrller tuning. Nevertheless, its accurate results prvide a basis fr evaluating the tuning techniques which fllw. First-Order Plus Dead Time Apprximatin The simplest tuning technique which utilizes the integral criteria is based n a first-rder plus dead time apprximatin t the prcess reactin curve. The prcess reactin curve is the pen lp respnse f the heat exchanger utlet temperature r cntrlled variable t a step change in the steam pressure r manipulated variable. The first-rder plus dead time mdel is f the frm: P (s) s ' = s T. (2 i n ts+1 s that there are three parameters K, t, and 9 which must be determined. Determinatin f the mdel parameters is easily accmplished by the graphical technique illustrated in Figure 2.3 (1). The prcess gain is then the steady-state change in the utlet temperature divided by the change in the steam pressure. The prcess dead time is fund by cnstructing a line tangent t the curve at the pint f maximum slpe. The intersectin f this tangent with the abscissa gives the prcess dead time directly. The p r cess time cnstant is the time required t achieve 62.3% f the final change in the utlet temperature minus the prcess dead time. Fr a first-rder dead time prcess, tuning parameters have been determined which satisfy all three integral criteria fr bth

32 13 LO a a 8j<3 CD [ [NJ 1 LD- 3 O tj Q I D - O ID_ Figure 2.3. First-Order Apprximatin t the Heat Exchanger Prcess Reactin Curve Obtained by Making a 5 psi Change in the Steam Pressure.

33 PI and PID cntrllers and fr bth setpint and lad changes (6,1). These results have been presented in terras f the prcess gain, prcess dead time, and time cnstant. Optimal Cntrl Tuning Methd fr Lad Changes Anther methd is available fr tuning fr lad changes using the integral criteria. This methd relies n a secnd-rder mdel f the prcess reactin curve and applies ptimal cntrl thery fr slutin f the tuning prblem (7). A secnd-rder mdel f the prcess reactin curve is f the frm: S W = (2.12)?s<s> s2 + 25ui s + OJ2 s n n where: u)^ = natural frequency K = prcess gain = damping rati Thus, it is again necessary t determine three parameters. A l thugh there are several techniques fr apprximating secnd-rder systems, ne methd is superir since it is applicable t underdamped as well as verdamped prcesses (8). Using this technique it was fund that the heat exchanger was indeed an underdamped prcess represented by the parameters =.75, u)^ =.452, and K =.697. The ptimal cntrl methd is best illustrated by transfrming the secnd-rder equatin int state variable frm by letting:

34 15 Cntrller Figure 2.4a. Blck Diagram f Secnd-Order System and Disturbance, gq. v 3 x2(> 8(0)j R Cntrller u X = DX+Eu Figure 2.4b. Blck Diagram f System Transfrmed t Third Order with Disturbance as Initial Cnditin.

35 16 dt X 2 = de (2.13) The secnd-rder prcess can then be written as: X 1 ~ X 2 X = Kt2g - 2puj X - ui2x 2 n ^ n 2 n 1 (2.14) with initial cnditins: X x(0) = 0 X 2 (0) = 0 The blck diagram f Figure 2.4a shws this secnd-rder mdel, an unspecified type f feedback cntrller, and a prcess disturbance, V By prper transfrmatin the disturbance can be changed t an initial cnditin. Nte that: g = u + gq (2.15) and g ~ u (2.16) This last equatin can be added t the secnd-rder system t give: X 1 0 X. 0J_ -2Fu) Kt n ^ n n X 2 0 u (2.17) S S 1

36 17 r: X = D X + e u a third-rder prcess with initial cnditins: Xx(0) 0 x2(0) 0 (2.18) g(0) g0 as illustrated in Figure 2.4b. Frm ptimal cntrl thery, it is knwn that the ptimal cntrller fr a linear system such as this is a linear functin f the states, that is: u = + K 2X 2 + K 3g (2.19) Hwever, frm the system equatins: X2 ' nx l " 2 5 % X2 + Kl n 8 (2.20) Rearranging: X. X, 2 X, g = 2 ^ a. + K Km K id n n (2.21) Substituting int the cntrl equatin and integrating: u = G 1X 1 + G 2 J t X 1dt + G 3X l (2.22) where: 2 K3 G 1 = K 2 + K m n

37 Thus, fr lad upsets, the ptimal feedback cntrller fr the riginal secnd-rder system is a PID cntrller. The cnstants in this cntrl law are determined by slutin f the matrix Riccati equatin which minimizes a cst f the frm J *= J f [XT a X + H T R «]dt (2.23) t where Q. and R are weighting matrixes. Fr the prblem at hand and R are chsen s that: J = J (X2 + ru2)dt (2.24) e0 This cst is the sum f the ISE integral and a penalty fr rapid changes in the cntrl. Such a penalty is quite desirable, es pecially when the cntrlling medium is lw in cst. The ptimal clsed-lp cntrl is: u = K X = K XX X + K 2X 2 + K 3X 3 (2.25) w h e r e : K = - et P/r (2.26) and: P = - P D - DT P + P e et P/r - & (2.27) with

38 19 P( tf) = 0 (2.28) Slutin f these equatins fr a chsen value f r gives the ptimum tuning parameters fr the PID cntrller. Figure 2.5 shws the time respnse f the heat exchanger u t let temperature fr different values f r. By bserving the r e spnse fr varius values f r, the user can chse the set f parameters which best suits the requirements f the prcess. While this requires sme experimentatin with the prcess, it is necessary t allw fr the differences between the simple secnd-rder mdel and the actual high-rder heat exchanger. Mdificatin fr Setpint Changes It must be remembered that the ptimal cntrl settings were develped specifically fr lad changes. As mentined earlier, the dynamic behavir f the standard PID algrithm is different fr lad and setpint changes. Thus the ptimal cntrl settings result in a smewhat less than ptimal respnse fr setpint changes if the standard PID algrithm is used. A pssible slutin t this prblem wuld be t use a cntrller algrithm which minimizes this dynamic difference between setpint and lad changes. It was stated in cnnectin with Equatin 2,18 that the ptimal cntrl fr a linear system is a linear functin f the states. It was shwn with Equatin 2,21 that fr lad changes this results in a PID cntrller based nly n the prcess utput and nt the errr as with the usual algrithm. T handle setpint changes in a practical manner, hwever, the integral mde must be based n the errr signal. Therefre, a mdified

39 CM O O - O ,00 5, DO Time, Sec. 10*00 Figure 2.5a. Clsed-Lp Outlet Temperature Respnse t a Lad Change fr Different Values f r.

40 m O* CD Q O in SO Time, Sec. Figure 2.5b. Cntrller Output Respnse t a Lad Change fr Different Values f r.

41 PID algrithm is: By basing the prprtinal and derivative mdes n the prcess utput, the initial rapid reactin f the standard PID algrithm is eliminated. As a result, the ptimal tuning parameters fr lad changes and fr setpint changes shuld be nearly the same fr a chsen cst criterin. Of curse, the respnse t lad changes is the same fr either the standard r mdified PID algrithm. Cmparisn f Tuning Results In rder t assess the relative merits f the tuning techniques previusly discussed, each methd was applied t the tuning f the heat exchanger cntrl system. The results f the accurate simulatin serve as a basis fr determining hw clsely a tuning methd actually cmes t minimizing the chsen integral criterin. Tables 2.1 and 2.2 list the cmputatinal results fr lad change and setpint change tuning respectively. Included fr each mdel tested are the ptimal parameters and crrespnding cst f the integral criterin. The minimum IAE and ITAE respnses t a change in the inlet temperature f il t the heat exchanger are given in Figure 2.6. This plt indicates that use f the ITAE criterin results in larger deviatins frm the setpint than with the IAE criterin. The ITAE penalizes heavily fr deviatins late in time but at the expense f initially large versht.

42 23 TABLE 2.1 RESULTS OF INTEGRAL TUNING FOR LOAD CHANGES Mdel Criterin Cntrller Setting Cst <1) Exact IAE PI K c s= T (2) 1st Order IAE PI K c = T. = 1.73 X (3) Exact ITAE PI K c = T. _ 1.63 X (4) 1st Order ITAE PI K c = T i = 1.65 (5) Exact IAE PID K c T, _.029 l Td.621 (6) 1st Order IAE PID K c = T l Td =.222 (7) Exact ITAE PID K c = T i d _.158

43 TABLE 2.1 CONTINUED Mdel Criterin Cntrller Setting (8) 1st Order ITAE PID K = c Cst 6220 T ± = 1.18 Td =.231 (9) Optimal ITAE PID K c = 286 Cntrl T± = Td =.207

44 25 TABLE 2.2 RESULTS OF INTEGRAL TUNING FOR SETPOINT CHANGES Mdel Criterin Cntrller Setting (1) Exact IAE PI K c = 40.6 Cst 148 (2) 1st Order IAE PI T 1 = 27.2 Kc = 5.69 T i = (3) Exact ITAE PI K c = T. = 25,2 i (A) 1st Order ITAE PI K c = 4.89 T i = 4.08 (5) Exact IAE PID K c = L = 81.1 =.240 (6) 1st Order IAE PID K c = T i = 5.69 T d =.246 <7) Exact ITAE PID K c = T i = 81.6 T d =.240 (8) 1st Order ITAE PID K c = T i = 5.29 T d =.212

45 26 TABLE 2.2 CONTINUED Mdel Criterin Cntrller (9) Optimal Cntrl ITAE PID (10) Optimal Cntrl ITAE Mdified PID (11) Exact ITAE Mdified PID Setting K» 286 c T i = 1 *17 Td =.207 K = 286 c T i = X 17 Td =.207 K = 130 c T = 1.05 I T, =.167 Cst

46 27 CD C\J CD CD Outlet temperature change CD CD IAE ITAE O n 0. DO Figure 2.6, S. DO Time, Sec Optimal PI Respnses t Lad Change with IAE and ITAE Criteria. e. s

47 Figures 2.7, 2.8, 2.11, and 2.12 shw that tuning fr lad changes using the first-rder lag plus dead time apprximatin results in relatively pr respnses cmpared t the ptimal r e spnses. Fr a lad change, as in Figures 2.7 and 2.8, the respnse is nearly unstable and still quite sluggish. At the same time, the minimum IAE and ITAE respnses fr the same upsets are straight lines in cmparisn. The ptimal cntrl methd tuning parameters, n the ther hand, yield a very gd respnse t lad changes. The heat e x changer utlet temperature respnse t a lad change using ptimal cntrl parameters is pltted in Figure Clearly, the difference between the ptimal cntrl and ptimal ITAE respnses is very small. Fr setpint changes Figures 2.9, 2.10, and 2.13 shw that nce again first-rder plus dead time tuning yields cmparatively pr respnse. The ptimal IAE and ITAE respnses, hwever, rise rapidly t the new setpint and line ut with n nticeable scillatin. Figure 2.13 indicates that use f the ptimal cntrl methd tuning parameters with the standard PID algrithm gives a large initial versht which is quite different frm the ptimal ITAE respnse. Hwever, if the ptimal cntrl parameters are used with the mdified PID algrithm the setpint respnse is quite clse t the ptimal ITAE respnse fr this algrithm as shwn by Figure Nevertheless, entries (7) and (11) f Table 2.2 indicate that the ITAE cst is higher fr the mdified PID algrithm than

48 29 Outlet Temperature Change, F ,50 D.DC 0.50 IAE S. 0 0 Time, Sec. 7. SQ Figure 2.7. Clsed-Lp PI Respnses t Lad Change with Optimal IAE and lst-rder plus Dead Time Parameters.

49 30 Q CZJ ITAE.c LO ^ l s t-order. DO e. s a S.OCJ Time, Sec. 7. S O 10. Q O Figure 2,8. Clsed-Lp PI Respnses t Lad Change with Optimal ITAE and lst-order plus Dead Time Parameters.

50 31 Outlet Temperature Change, F DO Off IAE. a. a s. aa 7. sa Time, Sec. Figure 2.9. Clsed-Lp PI Respnses t Setpint Change with Optimal IAE and lst-order Plus Dead Time Parameters.

51 32 CD CD CD ITAE CD, tac AJ CD CD B cu lst-order r I JJ CD O Q cb. n Time, Sec. Figure Clsed-Lp PI Respnses t Setpint Change with Optimal ITAE and lst-order Plus Dead Time Parameters.

52 Outlet Temperature Change, F.50-1 *00-0*50 0,00 0,50 1,00 lst-order Time, Sec. Figure Clsed-Lp PID Respnses t Lad Change with Optimal IAE and lst-order Plus Dead Time Parameters.

53 CD CD ltd a CD CD Lr> _ XTAE Optimal Cntrl lst-order CD CD UT) a. a S s Time, Sec Figure Clsed Lp PID Respnses t Lad Changes With Optimal XTAE, lst-order Plus Dead Time, and Optimal Cntrl Parameters.

54 LT> O O Optimal Cntrl, r = 10 O cn Q OJ lst-order a , D O Time, Sec,. 5 0 Figure Clsed-Lp PID Respnses t Setpint Changes With Optimal ITAE, lst-order Plus Dead Time and Optimal Cntrl Parameters.

55 Lf) O Q a* Outlet Temperature Change, F CJ m j ITAE V 6 Optimal Cntrl, r = Time, Sec Figure 2,14. Clsed-Lp Respnses t Setpint Changes Using Mdified PID Algrithm With Optimal ITAE and Optimal Cntrl Methd Parameters.

56 fr the standard algrithm. This is due t the slwer initial respnse f the mdified algrithra--precisely what was intended. Based n these results it was cncluded that the best f the methds studied fr tuning such high-rder systems as a heat e x changer is the ptimal cntrl methd. Als the mdified PID algrithm gave a better respnse, i.e. a lwer cst, fr setpint changes than simply using the ptimal cntrl parameters with the standard PID algrithm. Summary This chapter has dealt with the use f simple mdels such as first-rder lag plus dead time and secnd-rder t tune the feedback temperature cntrller n a high-rder prcess such as a heat exchanger. Cmparisn with the results f an accurate heat exchanger simulatin shwed that fr lad changes a secnd-rder mdel cmbined with ptimal cntrl thery yielded the best results. By using a mdified PID algrithm with prprtinal and derivative mdes based n the prcess utput instead f the errr, the ptimal cntrl parameters gave a nearly ptimal setpint respnse. In Chapter III this same apprach f tuning high-rder systems using simple mdels will be fllwed again. Hwever, instead f cnsidering cntinuus cntrllers as in this chapter, attentin will be devted t the tuning f digital r sampled-data cntrllers.

57 NOMENCLATURE LAE ITAE gq K K c Ps Integral abslute errr Integral time abslute errr Lad Mdel gain Cntrller gain Steam pressure r T Optimal cntrl weighting factr Heat exchanger utlet temperature T, Cntrller derivative time T^ Cntrller integral time T Heat exchanger inlet temperature T g Steam temperature T^ Tube wall temperature u Manipulated variable $ Dead time 5 Damping rati t Time cnstant Natural frequency

58 LITERATURE CITED 1. Murrill, P. W., Autmatic Cntrl f Prcesses, Internatinal Textbk Cmpany, Scrantn, Pa. (1967). 2. Lpez, A. M., P. W. Murrill, and C. L. Smith, "Optimal Tuning f Prprtinal Digital Cntrllers", DDC Tuning Reference Bk, Instruments and Cntrl Systems Reference Bk 69-01, Rimbach Publicatins, Philadelphia, Pa., (1969). 3. Chen, W. C. and E. F. Jhnsn, "Distributed Parameter Prcess Dynamics", Prcess Dynamics and Cntrl, Chemical Engineering Prgress Sympsium Series, Vl. 57, N. 36, (1961), p Mre, C. F., C. L. Smith, and P. W. Murrill, Multidimensinal Optimizatin Using Pattern Search, IBM Share Library, PID N. 6248, (1968). 5. Kern, D, 0., Prcess Heat Transfer, McGraw-Hill Bk C., Inc., New Yrk, New Yrk, (1950), p Rvira, A. A., P. W. Murrill, C. L. Smith, "Tuning Cntrllers fr Setpint Changes", Instruments and Cntrl Systems, Vl. 4 2, N. 12, (December, 1969), p Jhnsn, C. D., "Optimal Cntrl f the Linear Regulatr with Cnstant Disturbances", IEEE Transactins n Autmatic Cntrl, (August, 1968), p Meyer, J. R., G. D. Whitehuse, C, L. Smith, and P. W, Murrill, "Simplifying Prcess Respnse Apprximatin", Instrument and Cntrl Systems, Vl. 40, N, 12, (December, 1967), p. 76.

59 CHAPTER III TUNING OF SAMPLED-DATA CONTROLLERS FOR HIGH-ORDER SYSTEMS Intrductin It is nly in the past few years that the digital cmputer has been used fr direct cntrl f industrial prcesses. The fact that direct digital cntrl is becming widespread is due t its ability t prvide better prcess cntrl than cnventinal analg cntrllers. The flexibility f digital prgramming allws the practical Implementatin f such advanced cntrl methds as feed-frward cntrl, nn-interacting cntrl, and mdel-referenced adaptive cntrl. Nevertheless, fr the majrity f cntrl lps in a plant, the tendency has been t simply use a digital PI r PID cntrller. Hwever, the sampling prcess inherent in digital cntrl can significantly affect the dynamics f the cntrl lp. As a result, techniques fr tuning cntinuus cntrllers cannt be applied un mdified t digital cntrllers. The difference between cntinuus and sampled-data cntrllers als raises a questin abut the basic effectiveness f the digital PI and PID algrithms cmpared t ther digital algrithms which might be frmulated. 40

60 As with cntinuus cntrller tuning, digital cntrller tuning methds are based n the behavir f simple mdels. This chapter deals with the applicatin f such methds t the tuning f a heat exchanger utlet temperature cntrller--a typical high-rder system. Attentin is given nt nly t digital PI and PID algrithms but als t ther digital algrithms based n simple prcess mdels. Sampled-Data Temperature Cntrl f a Heat Exchanger Once again, the prcess being cnsidered is the il-cndensing steam heat exchanger discussed in Chapter II. The utlet temperature is cntrlled, as befre, by adjusting a valve in the steam line as shwn in Figure 2.1a. Hwever, in this case, a sampled - data feedback cntrller is used as shwn in Figure 3.1. Very significant parts f the cntrl system are the cntrller utput sampler and the zer-rder hld which cnvert the discrete cntrller utput t a piece-wise cnstant signal prviding a cntinuus signal fr the cntrl valve. As mentined, the digital equivalent f the analg PI r PID algrithm is ften used fr feedback cntrl. The digital algrithm is btained by apprximating the integral and derivative mdes f the analg algrithm using numerical techniques. A simple trapezidal integratin is used fr the integral mde and a finite difference fr the derivative mde t give:

61 where: m^ = manipulated variable at the n*"*1 sampling instant = manual mde utput fch = errr signal at the n sampling instant T = sampling time Obviusly, as the sampling time appraches zer, the digital algrithm appraches the cntinuus PID algrithm in its behavir. It has been shwn that sampling causes a deteriratin in the cntrl perfrmance and that the digital PID algrithm at its very best perfrms as a cntinuus PID cntrller (1). Digital PI and PID Cntrller Tuning One methd f tuning digital PI and PID cntrllers is using an accurate simulatin f the heat exchanger and the feedback cntrller. Just as with cntinuus cntrllers in Chapter II, the prcedure Is t numerically slve the system equatins t btain the time respnse t a setpint r lad change. Using an ptimizatin rutine the cntrller parameters are varied t determine the set f parameters which minimize a chsen integral criterin-- in this study either IAE r ITAE. Fr a wrst case analysis, the upsets are intrduced int the prcess at a pint in time which is ne dead time befre a sampling instant. While nt a practical way t tune cntrllers in a plant, these accurate results prvide a basis fr cmparing the results f ther tuning methds which fllw.

62 A mre practical methd is an extensin f the technique f Chapter II which used a first-rder lag plus dead time mdel f the prcess reactin curve t tune cntinuus cntrllers. The methd is easily extended t sampled-data cntrllers by recgnizing that the sampler and the zer-rder hld are, in many cases, equivalent t a pure dead time f ne-half the sampling interval (2). Thus, the cntrller parameters are calculated using the prcess gain and time cnstant f the prcess reactin curve mdel and an equivalent dead time: 0e = 6 + T/2 (3.2) where: 0^ = equivalent dead time 0 = prcess dead time In a strict sense this is an apprximatin which imprves as t / T increases. Fr systems where t and T are the same rder f magnitude, it is qualitatively reasned that using the equivalent dead time: 9 = 8 + T (3.3) wuld result in better cntrl (3). Deadbeat Algrithm fr Setpint Changes Since the cnventinal PI and PID cntrl deterirates when cnverted t digital cntrl and since the flexibility f digital prgramming allws practical implementatin f nn-standard algrithms fr digital cntrl systems, attentin has been given t develping new algrithms. One such algrithm, the deadbeat algrithm fr

63 44 setpint changes, is based n the nw familiar first-rder lag plus dead time mdel (4). Referring t Figure 3.1, the prcess utput respnse t setpint changes can be written in Z-transfrm ntatin as: Ctrt D < O G hg p ( Q R(z) 1 + D(z)GhGp(z) M< ^ ^ Fr cntrller design this is rearranged t give: D(2) - P<i) (3-5) By inserting the apprpriate expressin fr G^G (z) and specifying the desired respnse t a setpint change, M ( z ), the required digital cntrl can be determined. With a first-rder lag plus dead time prcess and a zer-rder h l d : G, G (z) = Z h p -Ts -Os'! 1-e. Ke i rr (3.6) s t s +1 Fr the case where 0 is less than T: Kr2(l-e"(T'0)/Ti - (e't/t-e'(t"6)/tn GhG (.) - H - I L S > U (3.7) * z(z-e ) The design criteria fr the deadbeat algrithm are that the utput reach the new setpint in a minimum number f sampling perids and that the system have n errr at the sampling pints. With these requirements: R ( z ) = ; (3.8a)

64 C(2) Lad E(s) D(z) C(s) Cntrller Zer-Order Hld Prcess Figure 3.1. Schematic Diagram f Sampled-Data Feedback Cntrl System.

65 46 (3.8b) M(z) = z -1 (3.8c) Substitutin f Equatin 3.7 and 3.8c int Equatin 3.5 gives the deadbeat digital cntrller. Simplifying and taking the inverse Z-transfrm gives the discrete algrithm: (3.9) where: Digital Predictr Algrithm Still anther algrithm utilizes the apprximatin that a zer-rder hld can be treated as a pure dead time equal t ne- half the sampling time. The prcess utput at a future time is analytically predicted based n the respnse f a first-rder lag plus dead time mdel (5). The algrithm includes a prprtinal mde and, t quickly eliminate the effect f unmeasured disturbances, a numerical integral mde. The prcess is represented by a first-rder lag plus dead time mdel f the prcess reactin curve. In this case, hwever, a term is added t accunt fr unmeasured disturbances s that: c(s) = M s ) + d ( s ) T K e ~63 TS + 1 (3.10)

66 w h ere: d = unmeasured disturbance The cntrl law fr this algrithm is a prprtinal actin,based n the difference between the actual prcess utput and the a n al y tically predicted utput at a time 0 + T/2 in the future, plus a calculated disturbance term t accunt fr unmeasured disturbances, i.e. : nt+0+t/2^ " dnt (3.11) where: u = cntrl actin at time nt nt d m = calculated disturbance at time nt ni K = cntrller gain R = calibrated cntrller setpint XnT+0+T/2 = analytically predicted prcess utput at time nt+e+t/2 n = an integer In this way the effects f the transprtatin lags assciated with the prcess and the sample and hld are eliminated. The predicted prcess utput is determined by integrating the mdel equatin and evaluating the result ver tw separate intervals. As illustrated in Figure 3.2 fr the case where 0 is less than T, the utput at 0 depends n the cntrl actin, and the calculated disturbance, d, s that:

67 + T/2 Prcess Output -1 Cntrl -2 Calculated Disturbance -2T 0 Figure 3.2. Prcess Output, Cntrl Actin, and Calculated Disturbance f Digital Predictr Algrithm.

68 49 X Q = K( 1 - C ') (u_1+ d Q) + C'X (3.12) where: x 0 = prcess utput at a time 0 in the future x q «= prcess utput at time equal zer C' = e_0/t The prcess utput at 0 + T/2 depends n the cntrl actin, u q, and again the calculated disturbance, d^, giving: V i / 2 = K <1 -A '>(V d> + A 'X S <3 -l3> where: A ' = e T/2T Eliminating x. frm these equatins yields the predicted prcess 0 utput: X 6+T/2 = K (l'a')(u+d0) + A ' K C l - C ' K u. j + d ^ + A'C'Xq (3.14) Cmbining Equatins 3.11 and 3.14 and slving fr the cntrl actin gives the digital predictr algrithm: UnT = C 1SP " C 2X nt " C3 (unt-l+dnt') " dnt (3.15) w h e r e : C1 = (KcKH4)/K[l+KcK(l-A')] c, =ka'c7[i+kk(i-a')] fc C C C 3 = KcK A ' ( l - C O / [ H - K ck(l-a');i

69 SP = setpint Nte that a setpint calibratin has been included t frce the clsed-lp gain frm prcess setpint t prcess utput t equal ne. The disturbance is calculated using the numerical integral mde given by: dnt = dnt-l + K r(xnt~x n P T (3.16) where: X nl = analytically predicted utput at time nt K = reset gain r The calculated utput at the sampling pint is fund frm the analytical slutin f the mdel equatin fr the preceeding sampling interval. This utput is: X q = KCl-B'/C'Xu.j+d.j) + (GB// C /)(l.-c')(ll_2+ d -l) (3.17) where: Obviusly, with perfect mdeling, this integral mde is active nly when the prcess is upset by an unmeasured disturbance and nt when setpint changes are made. The values f and are calculated by cnsidering the clsed-lp respnse f a first-rder prcess being cntrlled by the predictr algrithm. Requiring a minimal r deadbeat respnse i

70 fr bth setpint and lad changes results in: Kc K(l-A) (3.18) Experience in applying the algrithm t prcesses which are nt first-rder has indicated that the value f is t large and that using.1k^ results in a better respnse fr bth setpint and lad changes. Cmparisn f Tuning Results In rder t cmpare the effectiveness f the tuning methds cnsidered in this chapter, each technique was applied t tuning the heat exchanger cntrl system using a sampling time f tw secnds. The results f the accurate simulatin serve as a basis fr determining hw clsely the PI and PID tuning methds cme t minimizing a chsen integral criteria. These results als prvide a means f cmparing the effectiveness f the deadbeat, predictr, PI, and PID algrithms. Tables 3.1 and 3.2 list the results fr each tuning methd fr lad and setpint change tuning, respectively. Included fr each mdel are the cntrller settings and the integral cst. Figures 3.3 thrugh 3.6 illustrate the lad and setpint change respnses f the heat exchanger with a digital PI cntrller. I n cluded are the results f ptimal tuning and first-rder lag plus dead time tuning using and equivalent dead time. These figures shw that using: